Nonlinear elliptic-parabolic dynamical boundary value problems without initial conditions (in Ukrainian)

Yu.B.Dmytryshyn

We investigate a nonlinear elliptic-parabolic dynamical boundary value problem without initial conditions. Sufficient conditions for the existence and uniqueness of the generalized solution are obtained without an additional assumption on the behavior of the solution and data-in at $t\to-\infty$. Furthermore, the continuous dependence of the generalized solution on the data-in is proved.

1. Showalter~R.~E. Monotone operators in Banach space and nonlinear partial differential equations, volume 49 of Mathematical Surveys and Monographs. -~Providence: Amer. Math. Soc.,~1997. - 278~p.

2. Friedman~A., Shinbort~M. The initial value problem for the lianerized equations of water-waves // J.~Math. Mech. -~1967. -~V.17. -~P.~107-180.

3. Garipov~R.~M. On the linear theory of gravity waves: the theorem of existence and uniqueness // Archive Rat. Mech. Anal. -~1967. -~V.24. -~P.~352-362.

4. Bejenaru~I.,~D\'az~J.~I., Vrabie~I.~I. An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions // Electron. J. Differential Equations. -~2001. -~V.2001,~№50. -~P.1-19.

5. Vitillaro~E. On the Laplace equation with nonlinear dynamical boundary conditions // Proc. London Math. Soc. -~2006. -~V.93, №3. -~P.418-446.

6. Showalter~R.~E. Nonlinear degenerate evolution equations and partial differential equations of mixed type // SIAM J. Math. Anal. -~1975. -~V.6, №1. -~P.25-42.

7. Showalter~R.~E. Existence and representation theorems for a semilinear Sobolev equation in Banach space // SIAM J. Math. Anal. -~1972. -~V.3. -~№3. -~P.~527-543.

8. Hintermann~T. Evolution equations with dynamic boundary conditions // Proc. Roy. Soc. Edinburgh. -~1989. -~V.113A, №1/2. -~P.43-60.

9. Grobbelaar-Van Dalsen~M. On $B$-evolution theory and dynamic boundary conditions on a portion of the boundary // Appl. Anal. -~1991. -~V.40, №2/3. -~P.151-172.

10. Escher~J. Nonlinear elliptic systems with dynamic boundary conditions // Math. Z. -~1992. -~V.210, №3. -~P.413-439.

11. Escher~J. Quasilinear parabolic systems with dynamical boundary conditions // Commun. Partial Differ. Equations. -~1993. -~V.18, №7-8. -~P.1309-1364.

12. Arrieta~J.~M., Quittner~P., Rodr\'guez-Bernal~A. Parabolic problems with nonlinear dynamical boundary conditions and singular initial data // Differential Integral Equations. -~2001. -~V.14, №12. -~P.1487-1510.

13. Di Benedetto~E., Showalter~R.~E. Implicit degenerate evolution equations and applications // SIAM J. Math. Anal. -~1981. -~V.12, №5. -~P.731-751.

14. Igbida~N., Kirane~M. A degenerate diffusion problem with dynamical boundary conditions // Math. Ann. -~2002. -~V.323, №2. -~P.377-396.

15. Andreu~F., Igbida~N.,~Maz\'on~J.~M. Toledo~J. A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions // Interfaces Free Bound. -~2006. -~V.8, №4. -~P.447-479.

16. Лионс~Ж.-Л. Некоторые методы решения нелинейных краевых задач. -~М.: Мир,~1972. -~587~с.

17. Amann~H., Fila~M. A Fujita-type theorem for the Laplace equation with a dynamical boundary condition // Acta Math. Univ. Comenianae. -~1997. -~V.66, №2. - P.321-328.

18. Fila~M., Quittner~P. Global solutions of the Laplace equation with a nonlinear dynamical boundary condition // Math. Methods Appl. Sci. -~1997. -~V.20, №15. -~P.1325-1333.

19. Vitillaro~E. Global existence for the heat equation with nonlinear dynamical boundary conditions // Proc. Roy. Soc. Edinburgh. -~2005. -~V.135A, №1. -~P.175-207.

20. Бокало~М.~М., Дмитришин~Ю.~Б. Нелінійна динамічна крайова задача без початкової умови для квазілінійних еліптичних рівнянь // Нелинейные граничные задачи. -~2007. - Т.17. - С.1-19.

21. Bokalo~M., Dmytryshyn~Yu. Problems without initial conditions for degenerate implicit evolution equations // Electron. J. Differential Equations. -~2008. -~V.2008, №4. -~P.1-16.

22. Лионс~Ж.-Л., Мадженес~Э. Неоднородные граничные задачи и их приложения. -~М.: Мир. -~1971. -~371~с.

23. Adams~R.~A. Sobolev spaces. -~New York, San Francisco, London: Academic Press, ~1975. -~xviii+270~p.

24. Гаевский~Х.,~Грёгер~К. Захариас~К. Нелинейные операторные уравнения и операторные дифференциальные уравнения. -~М.: Мир,~1978. -~336~с.

25. Данфорд~Н., Шварц~Дж. Линейные операторы. Т. 1: Общая теория. -~М.: Изд. ИЛ,~1962. -~898~с.

26. Showalter~R.~E. Hilbert space methods for partial differential equations. -~Pitman, London-San Francisco, Calif.-Melbourne, ~1977. - 219~p.

	Nonlinear elliptic-parabolic dynamical boundary value problems without initial conditions (in Ukrainian)
Author	Yu.B.Dmytryshyn yuree@yandex.ru Ivan Franko National University of Lviv
Abstract	We investigate a nonlinear elliptic-parabolic dynamical boundary value problem without initial conditions. Sufficient conditions for the existence and uniqueness of the generalized solution are obtained without an additional assumption on the behavior of the solution and data-in at $t\to-\infty$. Furthermore, the continuous dependence of the generalized solution on the data-in is proved.
Keywords	nonlinear elliptic-parabolic equation, dynamical boudary value problem, generalized solution
DOI	doi:10.30970/ms.32.1.53-63
Reference	1. Showalter~R.~E. Monotone operators in Banach space and nonlinear partial differential equations, volume 49 of Mathematical Surveys and Monographs. -~Providence: Amer. Math. Soc.,~1997. - 278~p. 2. Friedman~A., Shinbort~M. The initial value problem for the lianerized equations of water-waves // J.~Math. Mech. -~1967. -~V.17. -~P.~107-180. 3. Garipov~R.~M. On the linear theory of gravity waves: the theorem of existence and uniqueness // Archive Rat. Mech. Anal. -~1967. -~V.24. -~P.~352-362. 4. Bejenaru~I.,~D\'az~J.~I., Vrabie~I.~I. An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions // Electron. J. Differential Equations. -~2001. -~V.2001,~№50. -~P.1-19. 5. Vitillaro~E. On the Laplace equation with nonlinear dynamical boundary conditions // Proc. London Math. Soc. -~2006. -~V.93, №3. -~P.418-446. 6. Showalter~R.~E. Nonlinear degenerate evolution equations and partial differential equations of mixed type // SIAM J. Math. Anal. -~1975. -~V.6, №1. -~P.25-42. 7. Showalter~R.~E. Existence and representation theorems for a semilinear Sobolev equation in Banach space // SIAM J. Math. Anal. -~1972. -~V.3. -~№3. -~P.~527-543. 8. Hintermann~T. Evolution equations with dynamic boundary conditions // Proc. Roy. Soc. Edinburgh. -~1989. -~V.113A, №1/2. -~P.43-60. 9. Grobbelaar-Van Dalsen~M. On $B$-evolution theory and dynamic boundary conditions on a portion of the boundary // Appl. Anal. -~1991. -~V.40, №2/3. -~P.151-172. 10. Escher~J. Nonlinear elliptic systems with dynamic boundary conditions // Math. Z. -~1992. -~V.210, №3. -~P.413-439. 11. Escher~J. Quasilinear parabolic systems with dynamical boundary conditions // Commun. Partial Differ. Equations. -~1993. -~V.18, №7-8. -~P.1309-1364. 12. Arrieta~J.~M., Quittner~P., Rodr\'guez-Bernal~A. Parabolic problems with nonlinear dynamical boundary conditions and singular initial data // Differential Integral Equations. -~2001. -~V.14, №12. -~P.1487-1510. 13. Di Benedetto~E., Showalter~R.~E. Implicit degenerate evolution equations and applications // SIAM J. Math. Anal. -~1981. -~V.12, №5. -~P.731-751. 14. Igbida~N., Kirane~M. A degenerate diffusion problem with dynamical boundary conditions // Math. Ann. -~2002. -~V.323, №2. -~P.377-396. 15. Andreu~F., Igbida~N.,~Maz\'on~J.~M. Toledo~J. A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions // Interfaces Free Bound. -~2006. -~V.8, №4. -~P.447-479. 16. Лионс~Ж.-Л. Некоторые методы решения нелинейных краевых задач. -~М.: Мир,~1972. -~587~с. 17. Amann~H., Fila~M. A Fujita-type theorem for the Laplace equation with a dynamical boundary condition // Acta Math. Univ. Comenianae. -~1997. -~V.66, №2. - P.321-328. 18. Fila~M., Quittner~P. Global solutions of the Laplace equation with a nonlinear dynamical boundary condition // Math. Methods Appl. Sci. -~1997. -~V.20, №15. -~P.1325-1333. 19. Vitillaro~E. Global existence for the heat equation with nonlinear dynamical boundary conditions // Proc. Roy. Soc. Edinburgh. -~2005. -~V.135A, №1. -~P.175-207. 20. Бокало~М.~М., Дмитришин~Ю.~Б. Нелінійна динамічна крайова задача без початкової умови для квазілінійних еліптичних рівнянь // Нелинейные граничные задачи. -~2007. - Т.17. - С.1-19. 21. Bokalo~M., Dmytryshyn~Yu. Problems without initial conditions for degenerate implicit evolution equations // Electron. J. Differential Equations. -~2008. -~V.2008, №4. -~P.1-16. 22. Лионс~Ж.-Л., Мадженес~Э. Неоднородные граничные задачи и их приложения. -~М.: Мир. -~1971. -~371~с. 23. Adams~R.~A. Sobolev spaces. -~New York, San Francisco, London: Academic Press, ~1975. -~xviii+270~p. 24. Гаевский~Х.,~Грёгер~К. Захариас~К. Нелинейные операторные уравнения и операторные дифференциальные уравнения. -~М.: Мир,~1978. -~336~с. 25. Данфорд~Н., Шварц~Дж. Линейные операторы. Т. 1: Общая теория. -~М.: Изд. ИЛ,~1962. -~898~с. 26. Showalter~R.~E. Hilbert space methods for partial differential equations. -~Pitman, London-San Francisco, Calif.-Melbourne, ~1977. - 219~p.
Pages	53-63
Volume	32
Issue	1
Year	2009
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html